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Claim. For every $m\ge1$, every set $A$ of $m$ distinct positive integers has at least $m^{1/2}$ distinct ratios $a/\gcd(a,b)$ with $a,b\in A$, and there are $m$-sets with at most $(3/2+o(1))(2m)^{2/3}$ such ratios: in the notation of Problem 539, $m^{1/2}\le h(m)\lesssim(3/2)(2m)^{2/3}$. This is Theorem 2 (p. 3) of A. Granville and F. Roesler, The set of differences of a given set, Amer. Math. Monthly 106 (1999), no. 4, 338--344, cited as [GrRo99] on the problem page, stated there for the vector form of the problem: with $\delta(\mathbf a,\mathbf b)=(\max\{0,a_i-b_i\})_i$ on exponent vectors, $h(m)$ is the least $|\delta(A)|$ over $m$-sets of distinct vectors with nonnegative integer entries. The lower bound is the pairing argument of p. 2: for fixed $\mathbf a$ the pairs $(\delta(\mathbf a,\mathbf b),\delta(\mathbf b,\mathbf a))$, $\mathbf b\in A$, are distinct, since $\mathbf b=\mathbf a-\delta(\mathbf a,\mathbf b)+\delta(\mathbf b,\mathbf a)$, so one of the two coordinate sets has at least $m^{1/2}$ values. The upper bound is the count for the sets $\{(x,y)\in\mathbb Z^2:x,y\ge0,\ L<x+y\le U\}$ with $L,U=((2m)^{2/3}\mp(2m)^{1/3})/2+O(1)$ (pp. 2--3), which the paper credits to Freiman and Lev without a reference; the site's commentary and Erdős's 1973 survey credit the lower bound, and the weaker upper bound $h(n)<n^{1-c}$, to Erdős and Szemerédi, who published no proof. Library home granville_1999_set_differences_given_set; result pages Unsolved problem, Theorem 1 and Theorem 2. The page numbers are those of the authors' eight-page preprint, public at https://dms.umontreal.ca/~andrew/PDF/Roesler.pdf, whose labels the journal version may not share.

Covers. The bounds $n^{1/2}\ll h(n)\ll n^{2/3}$ alone, with the explicit constants above. Not covered: the order of $h(n)$, which the problem asks to estimate; the exponent, which the 2026 result on the ProofCouncil claim page puts at $1/2$; and whether the lower bound is sharp in order, which Kitamura's claim page answers in the negative without a paper. Theorem 1 of the paper, the bound $(m/2)^{2/3}$ for sets built from two primes, concerns a restricted class of sets and settles no instance of the question.

Depends on. No page of this wiki: the pairing argument is complete as printed, and the count for the Freiman--Lev sets is the paper's own.

Acceptance. Refereed: the paper is a journal publication in the American Mathematical Monthly, volume 106, issue 4 (April 1999), the refereed evidence; the issue carries no day, so this page is dated to the first day of that month. The site's curator, Thomas Bloom, credits the two bounds to Erdős and Szemerédi and to Freiman and Lev and points to this paper for their proofs, but the site labels the problem OPEN, so that credit is not reviewed evidence. The paper's statements are checked against the text; the proof of Theorem 1 (p. 4) and the Freiman--Lev count are not checked by this corpus, and nothing is independently reviewed by this project.